# Regularity results on the flow map of periodic dispersive Burgers type equations and the Gravity-Capillary equations

Published in *arXiv preprint*, 2021

In the first part of this paper we prove that the flow associated to the Burgers equation with a non local term of the form $D^{\alpha-1} \partial_x u$, $\alpha \in [1,+\infty[$ is Lipschitz from bounded sets of $H^s_0({\mathbb T};{\mathbb R})$ to $C^0([0,T],H^{s-(2-\alpha)^+}_0({\mathbb T};{\mathbb R}))$ for $T>0$ and $s>1+\frac{2-\alpha}{\alpha-1}+\frac{1}{2}$, where $H^s_0$ are the Sobolev spaces of function with $0$ mean value, proving that the result obtained previously by the author is optimal on the torus. The proof relies on a pseudodifferential generalization of a complex Cole-Hopf gauge transformation introduced by T.Tao for the Benjamin-Ono equation.

In the second part of this paper we use a paralinearization version of the previous method to prove that a re-normalization of the flow of the one dimensional periodic gravity capillary equation is Lipschitz from bounded sets of $H^s$ to $C^0([0,T],H^{s-\frac{1}{2}})$ for $T>0$ and $s>3+\frac{1}{2}$. This proves that the result obtained previously by the author is optimal for the water waves system